Chapter 24 - Valn of asset classes & portfolios

[fischer]

Hi
I did not quite follow the bit in Ch 24, on page 13 (section on Consistency of bases - Interest rate). Could someone please help me along the lines of the example below if possible?
Suppose a Life ins co has bought n indentical bonds with term 3 years to back a contract (liability) it has sold to a PH. The total income from the bonds is £5000 pa.

Under the contract, the PH receives £5000 pa for a period of 3 years.

Following on from the 1st 2 lines on page 13-
Asset value = market value of the bonds (n*price of each bond).
Liability value = 5000/(1+i1) + 5000/(1+i2)^2 + 5000/(1+i3)^3

where in is the n-year spot rate (say obtained from the BoE website), n going from 1 to 3.
Q1 Is my understanding of the 1st 2 lines of page 13 correct?

Following on from the next 3 lines on page 13-
Asset value = 5000/(1+i1) + 5000/(1+i2)^2 + 5000/(1+i3)^3
Liability value = 5000/(1+i1) + 5000/(1+i2)^2 + 5000/(1+i3)^3

where in is the n-year spot rate (say obtained from the BoE website), n going from 1 to 3.
Q2 Is my understanding of the next 3 lines of page 13 correct?

Any help would be much appreciated.

 

[Anna Bishop]

Hi Fischer

I wonder if your example would work better with longer-term liabilities as it is then that I think you would observe more of a difference between a market-related discount rate and a long-term expected return.

Suppose that, 3 years' ago, a life insurer took on a one-off 20-year liability of £100,000, which it matched from outset with a 20-year zero-coupon bond. When the bond is purchased, its long-term expected return (or GRY) is 5%.

Using the discounted cashflow method (where the same discount rate is used for both A and L).

Value of assets = £100,000 / (1+i)^17 at i = 5%
Value of liabilities = £100,000 / (1+i)^17 at i = 5%

Using the market value apporach.

Value of assets = market value
Value of liabilities = £100,000 / (1+i)^17 at i = market-related discount rate

The market-related discount rate would be determined by looking at the current GRY on a 17-year ZCB. This could well have shifted from the 5% yield at which the bond was purchased. The current yield could be 4%, in which case i = 4%.

Another example would be for liabilities backed with equities. A life company's long-term view for the expected return on equities might be 7%.

Using the discounted cashflow method, both assets and liabilities would be valued using a discount rate of 7%. On the asset side, dividends would be projected using a dividend growth rate of g and discounted at 7%. On the liabilities side, the cashflows would be projected and then discounted at 7%.

Using the market value approach, the assets would be valued at market value. The liability cashflows would be projected and then discounted at a market-related rate. This market-related rate could be determined as the current GRY on a government bond of a suitably matcing duration to the liabilities + an equity risk premium.

 

[Anna Bishop]

Hi Fischer

I wonder if your example would work better with longer-term liabilities as it is then that I think you would observe more of a difference between a market-related discount rate and a long-term expected return.

Suppose that, 3 years' ago, a life insurer took on a one-off 20-year liability of £100,000, which it matched from outset with a 20-year zero-coupon bond. The company's long-term expected return on bonds is 5%. This may be based on what the GRY on the bond was at purchase or some other analysis that the company has done.

Using the discounted cashflow method (where the same discount rate is used for both A and L):

Value of assets = £100,000 / (1+i)^17 at i = 5%
Value of liabilities = £100,000 / (1+i)^17 at i = 5%

Using the market value apporach:

Value of assets = market value
Value of liabilities = £100,000 / (1+i)^17 at i = market-related discount rate

The market-related discount rate would be determined by looking at the current GRY on a 17-year ZCB. The current GRY will reflect all sorts of things, including current market sentiment and could well be different from 5%. Let's say the current GRY on 17-year bonds is 4%, in which case i = 4%.

Another example would be for liabilities backed with equities. A life company's long-term view for the expected return on equities might be 7%.

Using the discounted cashflow method, both assets and liabilities would be valued using a discount rate of 7%. On the asset side, dividends would be projected using a dividend growth rate of g and discounted at 7%. On the liabilities side, the cashflows would be projected and then discounted at 7%.

Using the market value approach, the assets would be valued at market value. The liability cashflows would be projected and then discounted at a market-related rate. This market-related rate could be determined as the current GRY on a government bond of a suitably matcing duration to the liabilities + an equity risk premium.

Anna